Question

which statement describes the graph of $f(x) = 4x^7 + 40x^6 + 100x^5$?○ the graph crosses the x-axis at $x = 0$ and touches the x-axis at $x = 5$.○ the graph touches the x-axis at $x = 0$ and crosses the x-axis at $x = 5$.○ the graph crosses the x-axis at $x = 0$ and touches the x-axis at $x = -5$.○ the graph touches the x-axis at $x = 0$ and crosses the x-axis at $x = -5$.

Explanation:

Step1: Factor the polynomial

First, factor out the greatest common factor $4x^5$:
$f(x)=4x^5(x^2 + 10x + 25)$
Then factor the quadratic:
$f(x)=4x^5(x+5)^2$

Step2: Identify roots and multiplicities

• For root $x=0$: multiplicity is 5 (odd).
• For root $x=-5$: multiplicity is 2 (even).

Step3: Relate multiplicity to x-axis behavior

• Odd multiplicity: graph crosses x-axis.
• Even multiplicity: graph touches x-axis.

Answer:

The graph crosses the x-axis at x = 0 and touches the x-axis at x = -5.